Op ti cal pro per ties of Cu(In1-xGax)5Se8 from ellip so me tric mea su re ments

Op ti cal pro per ties of Cu(In1-xGax)5Se8

from ellip so me tric mea su re ments*

Larissa Du rán1**, Jaime Cas tro1, Elvis Her nán dez1, Ángel Mu ñoz2,José Na ran jo3 and Carlos Alberto Du ran te Rin cón1

1La bo ra to rio de Cien cia de Ma te ria les, De par ta men to de Fí si ca, Fa cul tad Ex pe ri men tal. de Cien cias,Uni ver si dad del Zu lia, Apar ta do Pos tal 526, Ma ra cai bo, Ve ne zue la. 2La bo ra to rio de As tro no mía y Fí si ca

Teó ri ca (LAFT). De par ta men to de Fí si ca. Fa cul tad Ex pe ri men tal de Cien cias, Uni ver si dad del Zu lia,Ma ra cai bo, Ve ne zue la. 3Ins ti tu to Uni ver si ta rio de Tec no lo gía de Ma ra cai bo. Ma ra cai bo, Ve ne zue la.

Recibido: 30-11-05 Ac eptado: 10-04-06

Abs tractSpec tros co pic ellip so me try mea su re ments at room tem pe ra tu re were done on the sys tem

Cu(In1-xGax)5Se8. This allowed to de ter mi ne the real and ima gi nary parts of the com plex re frac ti vein dex N(hn), the real and ima gi nary parts of the com plex die lec tric func tion e(hn), the ab sorptioncoe ffi cient a(hn) and the re flec ti vity R(hn) for each stu died com po si tion. The va lue of the energygap Eg for each com po si tion was ob tai ned from fi tting the nu me ri ca lly ob tai ned se cond de ri va ti vespec tra of the ex pe ri men tal data e(hn), d2

e(hn)/d(hn)2, to analytic cri ti cal- point line sha pes. Theenergy gap va lues of CuIn5Se8 and CuGa5Se8, thus ob tai ned, are in good agre e ment with those re -por ted by other authors. The va lue of the high fre quency die lec tric cons tant e¥ for each com po si -tion was de ter mi ned fi tting the real re frac ti ve in dex n to the Sellmeier dis per sion for mu la.

Key words: Ellip so me try; op ti cal pro per ties; or de red va cancy se mi con duc tors.

Propiedades opticas de Cu(In1-xGax)5Se8 a partirde medidas elipsométricas

Re su menMe di das de elip so me tría es pectros cópica a tem pe ra tu ra am bien te se lle va ron a cabo so bre

el sis te ma Cu(In1-xGax)5Se8. Esto per mi tió de ter mi nar las par tes real e ima gi na ria del ín di ce dere frac ción com ple jo N(hn), las par tes real e ima gi na ria de de la fun ción die léc tri ca com ple ja e(hn),el coe fi cien te de ab sor ción a(hn) y la re flec ti vi dad R(hn) para cada com po si ción es tu dia da. El va -lor de la bre cha de ener gía Eg para cada com po si ción se ob tu vo ajus tan do los es pec tros de la se -gun da de ri va da de la data ex pe ri men tal e(hn), d2

e(hn)/d(hn)2, nu mé ri ca men te ob te ni dos, a cur -vas ana lí ti cas con pun tos crí ti cos. Los va lo res de la bre cha de ener gía de CuIn5Se8 y CuGa5Se8,así cal cu la dos, es tán en buen acuer do con los re por ta dos por otros au to res. El va lor de la cons -tan te die léc tri ca de alta fre cuen cia e¥ se ob tu vo para cada com po si ción ajus tan do el ín di ce dere frac ción real n a la for mu la de dis per sión de Sellmeier.

Pa la bras cla ve: Elip so me tría; pro pie da des óp ti cas; se mi con duc to res con va can tesor de na das.

Scien ti fic Jour nal of the Ex pe ri men tal Fa cul ty of Scien ces,at the Universidad del Zulia Vo lu me 15 Nº 2, April-June 2007

** Au tor para la co rres pon den cia. Tel.: + 58- 261- 7598160. E-mail: [email protected]

CIENCIA 15(2), 234 - 241, 2007Maracaibo, Venezuela

* Tra ba jo pre sen ta do en el V Con gre so de la So cie dad Ve ne zo la na de Fí si ca, Uni ver si dad del Zu lia.Nu cleo Pun to Fijo - Edo. Fal cón, Ve ne zue la, No viem bre 2005.

In tro duc tion

In for ma tion about spec tral de pend -ence of op ti cal pa rame ters such as di elec tric con stant, re frac tive in dex, re flec tiv ity andab sorp tion co ef fi cient are es sen tial in thechar ac teri za tion of ma te ri als that are usedin the fab ri ca tion of opto- electronic de vicesand also in the op ti mi za tion of the ef ficiencyof thin films so lar cells. One of the non- destructive tech niques that are be ing em -ployed lately for the op ti cal char ac teri za tionis spec tro scopic el lip so me try (SE) (1-4).

The SE (5, 6) is an ex cel lent tech niquefor meas ur ing di elec tric func tions, sincethese can be ob tained with out the need ofKramers- Kronig trans for ma tions. Thistech nique meas ures the change in the po -lari za tion state of the light re flected from thesam ple’s sur face. In ad di tion, its high sur -face sen si tiv ity al lows sur face con di tions tobe as sessed and op ti mized in real time.

The com plex di elec tric func tion e e e= -r ij of the sam ple, where er and ei arethe real and imagi nary parts, is cal cu latedfrom the el lip so met ric data us ing the iso tropic two- phase model that in volves the me diumand the sam ple (1). e is re lated to the com plexre frac tive in dex N through the re la tion e= N2=(n – jk)2. In this re la tion, n is the real part of the re frac tive in dex and k the ex tinc tion co ef fi -cient. These pa rame ters can be de ter mined by solv ing the fol low ing equa tions:

er n k= -2 2 [1]

ei nk= 2 [2]

The op ti cal ab sorp tion co ef fi cient a and the nor mal in ci dence re flec tiv ity R can becal cu lated through the re la tions:

ap

l=

4 k[3]

Rn k

n k=

- +

+ +

( )( )

11

2 2

2 2 [4]

To our knowl edge no SE data is avail -able on Cu(In1-xGax)5Se8 alloys, even thoughop ti cal studies have been per formed ear lier(7-10). In the pres ent work, us ing SE, we re -port on the spec tral de pend ence at roomtem pera ture, in the en ergy range be tween0.7 and 5.2 eV, of the ab sorp tion co ef fi cient,the com plex di elec tric func tion, the com plexre frac tive in dex, the re flec tiv ity and the highfre quency di elec tric con stant ofCu(In1-xGax)5Se8 al loys.

Ex pe ri men tal Methods

In gots of Cu(In1-xGax)5Se8 al loys withx=0, 0.2, 0.4, 0.6, 0.8 and 1.0 were pre paredby di rect fu sion of the stoi chio met ric mix -ture of con stitu ent ele ments of at least 5Npu rity in evacu ated quartz am poules. Thesewere pre vi ously coated with car bon. De tailsre lated to the crys tal growth have been re -ported ear lier (7, 8). The com po si tion of eachre sul tant al loy was de ter mined by atomicemis sion spec tros copy per formed on HNO3

so lu tions of the sam ples us ing an ICP Perkin El mer Op tima 3200 RL. As shown in Ta ble 1, this was close to the ideal theo reti cal value of the start ing com po si tion. As ob served by ather mal probe, sam ples cut from the in gotsshowed n- type con duc tiv ity for x£0.4 andp- type for x³0.6. The unit cell pa rame ters aand c for dif fer ent com po si tion x, ob tainedfrom the analy sis of x- ray pow der dif frac tiondata us ing a Sie mens D5005 dif frac to me terwith cop per an ode and Bragg- Brentano ge -ome try, are also given in Ta ble 1. These datafur ther con firm that a sin gle phase withtetrago nal struc ture oc curs for x³0.4,whereas for x<0.4, tetrago nal and hex ago nalphases co ex ist be low about 850°C (7).

El lipsomet ric meas ure ments weremade, us ing a spec tral el lip so me ter SO PRAES4G equipped with ro tat ing po lar izer, onsam ples that had sur face area of about 4mm2 and thick ness of around 1 mm. Tomini mize the con tri bu tion to the ex peri men -tal data of thin films of ox ide that are formedon the sam ple sur face due to oxi da tion andalso to re duce the ef fect of sur face rough -


L. Durán y et al. / Cien cia Vol. 15, Nº 2 (2007) 234 - 241 235

ness to which el lip so me try is very sen si tive,the sam ples were pol ished to at tain op ti calqual ity just be fore ac quir ing el lip so met ricdata. The fo cused light spot on the sam plesur face al lowed lo cat ing the ade quate pointto achieve the best re sponse.

Re sults and Dis cus sion

The varia tion of the re frac tive in dex nand the ex tinc tion co ef fi cient k as func tionsof wave length for the Cu(In1-xGax)5Se8 sam -ples are shown in Fig ures 1 and 2, re spec -tively. We ob served that the real in dex spec -tra have an over all simi lar be hav ior. This isalso valid for the imagi nary in dex spec tra. In the spec tral re gion of low ab sorp tion, ac cu -racy of meas ure ments with the used el lip so -me ter (with out a com pen sa tor) is known tobe rather poor [1, 3]: we con sider that meas -ured val ues of k<0.1 are too im pre cise andin ac cu rate. It is worth em pha siz ing that themeas ured val ues of n are not af fected by this prob lem.

In Fig ure 2, it is sig nifi cant that allsam ples dis play, in higher or lower de gree,ab sorp tion tails be low the fun da men tal gapedge. These tails re sult from sev eral causes.Be sides in trin sic con tri bu tions such as al -loy dis or der, other causes such as grainbounda ries in poly crys tal line ma te rial and

de via tion from stoi chio me try can give rise tosuch tails (1).

From n and k, op ti cal mag ni tudes ofin ter est are cal cu lated eas ily. The ab sorp -tion co ef fi cient and the normal- incidence Rwere cal cu lated us ing the re la tions [3] and[4]. As in other al loy sys tems (1, 3) the de -pend ence of the re frac tion in dex on com po -si tion at a fixed en ergy be low the band gapis a smooth curve. While n is quite de pend -ent on sur face prepa ra tion, the value of R ismuch less sen si tive to thin over- layers andis thus a help ful quan tity for char ac teri za -tion pur poses. A suit able work ing en ergy inthe pres ent case is that of the near- infrared(IR) He- Ne la ser line, l = 1.523 µm (0.814eV). The val ues R(x) cal cu lated from ourdata are given in Fig ura 3. Our data are well rep re sented by the best- fit pa rab ola.

R x x= - +022359 002603 001072 0972. . . ( . ) [5]

The good fit im plies that this is a validmethod to evalu ate the com po si tion of theseal loys. How ever, it is im por tant to re veal thatval ues of R to x= 0.4 and 0.8 could not beenfit ted, cause in the low en ergy range pres entan un typ ical be hav ior, show ing a slide de via -tion due to su per fi cial prob lems.

The di elec tric func tion of a semi con -duc tor is closely linked to its elec tronic band


236 Op ti cal pro per ties of Cu(In1-xGax)5Se8 from ellip so me tric mea su re ments

Ta ble 1

Stoi chio me tric com po si tion and unit cell latti ce pa ra me ters of Cu(In1-xGax)5Se8 allo ys.

Cu In Ga Se a(Å) c(Å)

CuIn5Se8 0.7 4.7 - 7.6 5.718(5)4.030(1)h

11.62(3)16.372(1)h

Cu(In0.8Ga0.2)5Se8 0.6 3.9 0.8 7.6 5.6909(8) 11.3988(3)

Cu(In0.6Ga0.4)5Se8 0.8 3.0 1.6 7.5 5.6384(9) 11.2745(9)

Cu(In0.4Ga0.6)5Se8 1.1 1.5 3.1 7.6 5.5809(4) 11.1336(7)

Cu(In0.2Ga0.8)5Se8 1.0 0.6 4.0 7.7 5.5262(7) 11.0361(7)

CuGa5Se8 1.0 – 4.6 8.0 5.4810(4) 10.937(1)Va lues in the pa renthe sis in di ca te stan dard de via tion in the last di git.hhe xa go nal struc tu re, g= 120°.

struc ture. The fea tures ob served in e w( ) inthe op ti cal range are re lated to in ter bandtran si tions origi nated by large or sin gu larval ues of the joint va lence and con duc tionden sity of states. These points of the bandstruc ture are also de noted as van Hove sin -gu larities or criti cal points (CP’s) (11, 12).Sev eral CP’s are ob served in the spec tra of

Figure 1. They can be bet ter re solved in theirnu meri cal sec ond de riva tives. We cal cu lated de riva tives us ing smooth ing poly no mi alsand ob tained criti cal point en er gies from fit -ting to stan dard ana lytic line shapes (11- 14).

The CP’s can be ana lyzed in terms ofthis stan dard ana lytic line shapes:

e w wf( ) ( )= - - -C Ae E jJ nG [6]

The critical- point pa rame ters, am pli -tude A, en ergy thresh old E, broad en ing Gand ex ci tonic phase f are de ter mined by fit -ting the nu meri cally ob tained second- derivative spec tra d d2 2e w w( ) / of the ex peri -men tal data e w( ) to the re la tions:

d

d

n n Ae E j

Ae E j

n

n

j n

j

2

2

2

2

1 0e w

w

w

w

f

f

( ) ( ) ( )

( )=

- - - -

- -

¹-

-

G

G = 0[7]

The ex po nent n has the value -1/2 forone- dimensional (1D), 0 for 2D and ½ for 3DCP’s. Dis crete ex ci tons with Lo rentz ian lineshape are rep re sented by n= -1.

The fun da men tal band gap tran si tionswere fit ted only to the real part spec tra, andthe rest of struc ture were not con sid ered in



1 2 3 4 52,2

2,4

2,6

2,8

3,0

3,2

3,4

CuGa5Se

8

CuInGa4Se

8

CuIn2Ga

3Se

8

CuIn3Ga

2Se

8

CuIn4GaSe

8

CuIn5Se

8

n

energy (eV)

Figure 1. Wa ve length de pen den ce of the re -

frac ti ve in dex n of the sam ples

Cu(In1-x

Gax)

5Se

8.

1 2 3 4 50,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

CuIn

5Se

8

CuIn

4GaSe

8

CuIn

3Ga

2Se

8

CuIn2Ga

3Se

8

CuInGa4Se

8

CuGa5Se

8

k

energy (eV)

Figure 2. Wavelength dependence of the

extinction coefficient k of the samples

Cu(In1-x

Gax)

5Se

8.

0,0 0,2 0,4 0,6 0,8 1,0

0,208

0,212

0,216

0,220

0,224

R = 0.22359 - 0.02603x + 0.01072x2 (0.97)

R

x

Figure 3. Reflectivity R of Cu(In1-x

Gax)

5Se

8

calculated at the near infrared

wavelength of 1.523 µm of the He-Ne

laser. R is well fitted as a quadratic

function of the alloys composition x.

this work be cause the band struc ture ofthese al loys have not been cal cu lated yet in -clu sive the edges. This rea son makes im pos -si ble to fig ure out the mean ing of the rest ofthe struc tures. Thus, we fit ted all com po si -tions fol low ing this scheme. For all the sam -ples, the band gap fea tures were best fit tedwith three- dimensional criti cal points, f= 0and A re sults posi tive, correspond ing ourpeaks to M1 CP’s (11). It is re ported thatwhen f= 0, it rep re sents a pure mini mum(14) and cor re sponds to a Lo rentz ian lineshape (12).

Fig ures 4-7 shows the second- derivative spec tra of the real part of e for four rep re sen ta tive com po si t ion of theCu(In1-xGax)5Se8 sys tem. Ta ble 2 lists the en -ergy gap val ues measured by el lip so me try(Ege) and by op ti cal ab sorp tion (EgT) (7) for all stud ied com po si tions. The Eg val ues as afunc tion of x are shown in Fig ure 8. We ob -served EgT < Ege for all com po si tions. It isalso ob served that both EgT and Ege for theend com po si tions are in good agree mentwith the val ues re ported by other authors (7, 8, 10, 15- 17).

The Eg vs. x data in Fig ure 8 can be fit -ted to the re la tions:

EgT= 1.17214 + 0.45179 x + 0.20536 x2 (0.99) [8]

Ege= 1.28064 + 0.58825 x + 0.11518 x2 (0.99) [9]

Also, we re port the di elec tric con stantat high fre quency e¥ for each com po si tion.They have been de ter mined fit ting the realre frac tive in dex to the Sell meier dis per sionformula at en er gies be low the gap (18). TheSell meier for mula is given by:

na

b=

+ -

-

1 12 2

2

( ) )l

l[10]

where l is the wave length and a = ¥e 1 2/ and

b are the con stants to be de ter mined by thefit ting pro cess. Ta ble 2 shows the val ues of e¥ ob tained by this method. Un likely, it isim pos si ble to com pare with other re sults be -

cause these val ues are not re ported un tilnow, so this is an ini tial pro posal about thesys tem.

The op ti cal ab sorp tion co ef fi cient ofCu(In1-xGax)5Se8 al loys for dif fer ent val ues ofx, cal cu lated from the equa tion [3], is shown



0,75 1,00 1,25 1,50 1,75 2,00

-16

-12

-8

-4

0

4

8

12

d2 ?

r/d

?2

(eV

-2)

energy (eV)

Figure 4. Second-derivative, d2e

r(w)/dw

2,of the

real part of e for CuIn5Se

8. Symbol

display experimental data and line the

results of the fit.

0,75 1,00 1,25 1,50 1,75 2,00

-20

-10

0

10

20

30

d2? r

/d?

2(e

V-2

)

energy (eV)

Figure 5. Second-derivative, d2e

r(w)/dw

2,of the

real part of e for Cu(In0.8

Ga0.2

)5Se

8.

Symbol display experimental data

and line the results of the fit.

as a func tion of pho ton en ergy in Fig ure 9. Itcan be ob served that a for CuIn5Se8 (x =0) isabout 104 cm at roughly 1.18 eV, in creasessoftly with in ci dent en ergy and is above8·105 cm at 5 eV. Such a high value aroundthe fun da men tal ab sorp tion edge re flectsgood qual ity of the sam ples used in the pres -ent work. Simi lar be hav ior but with slightlylower values of a can also be noted in Fig -ure 9 near the band edge for other mem bersof the sys tem that have higher x.

Con clu sions

Op ti cal prop er ties at room tem pera ture for the Cu(In1-xGax)5Se8 al loys sys tem, cov er -ing the whole com po si tion range, are pre -sented. Spe cial at ten tion has been paid tothe prob lem of mini miz ing sur face ef fects onthe meas ure ments of n and k, so that the re -ported val ues are char ac ter is tic of the bulk.



1,2 1,4 1,6 1,8 2,0 2,2-18

-9

0

9

d2 ?

r/d

?2

(eV

-2)

energy (eV)

Figure 6. Second-derivative, d2er(w)/dw2,of the

real part of e for Cu(In0.4

Ga0.6

)5Se

8.

Symbol display experimental data

and line the results of the fit.

1,50 1,75 2,00 2,25 2,50-10,0

-7,5

-5,0

-2,5

0,0

2,5

5,0

7,5

d2 ?

r/d

E2

(eV

-2)

energy (eV)

Figure 7. Second-derivative, d2er(w)/dw2,of the

real part of e for CuGa5Se

8. Symbol

display experimental data and line

the results of the fit.

Ta ble 2

Energy gaps of the sys tem Cu(In1-xGax)5Se8 mea su red by ellip so me try (Ege) and op ti cal ab sorption

(EgT) [7]. High fre quency die lec tric cons tant e¥ also shown.

x EgT (eV) Ege (eV) e¥

0 1.17 1.28 7.1893

0.2 1.28 1.41 7.1799

0.4 1.38 1.54 6.7660

0.6 1.50 1.64 6.9542

0.8 1.69 1.86 7.7981

1.0 1.82 1.97 6.8543

From the analy sis of the nu meri cal sec -ond de riva tives of the real part of e w( ), wehave ob tained the band gap en er gies foreach com po si tion. The en ergy gap val ues are in good agree ment with those re ported byother authors. The cali bra tion curve R(x) at l =1523. µm can be ap plied to de ter mine theal loy com po si tion, for in stance, in thin filmsof these ma te ri als, with all the ad van tages of op ti cal meth ods. We have re ported the di -elec tric con stant at high fre quency, e¥ , foreach com po si tion as an ini tial pro posal. Theop ti cal ab sorp tion co ef f i cient a ofCu(In1-xGax)5Se8 al loys with dif fer ent val ues

of x, cal cu lated from the el lip so met ric datahave shown a high value around the fun da -men tal ab sorp tion edge re flect ing good qual -ity of the sam ples used in the pres ent work.

Acknowledg ments

This work was sup ported by grantsfrom FONA CIT and CONDES- LUZ.

Re fe ren ces

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0,0 0,2 0,4 0,6 0,8 1,01,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9

2,0

Eg

(eV

)

x

Figure 8. Variation of the energy gap as a

function of composition for the

Cu(In1-x

Gax)

5Se

8 system. Open squares

represent data obtained from optical

absorption and full squares represent

data from ellipsometry mea su-

rements. The solid curves represent

the fit according to equations (8) and

(9).

0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0

104

105

106

Abs

orpt

ion

Coe

ffic

ient

(cm

-1)

hn

x = 0 x = 0.2 x = 0.4 x = 0.6 x = 0.8 x = 1

Figure 9. Optical absorption coefficient of

Cu(In1-x

Gax)

5Se

8 alloys as a function of

incident photon energy between

0.7 and 5.2 eV obtained from

ellipsometric data.

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Op ti cal pro per ties of Cu(In1-xGax)5Se8 from ellip so me tric mea su re ments